Abstract

A design procedure is described to determine the thicknesses of single-layer coatings of a given dielectric on a given metallic substrate so that a specified net phase retardance (and/or a net relative amplitude attenuation) between the p and s polarizations is achieved after three reflections from a symmetrical arrangement of three mirrors that maintain collinearity of the input and output beams. Examples are presented of halfwave and quarterwave retarders (HWR and QWR) that use a ZnS–Ag film–substrate system at the CO2-laser wavelength λ = 10.6 μm. The equal net reflectances for the p and s polarizations are computed and found to be high (above 90%) for most designs. Sensitivity of the designs (deviation of the magnitude and phase of the ratio of net complex p and s reflection coefficients from design specifications) to small film-thickness and angle-of-incidence errors is examined, and useful operation over a small wavelength range (10–11 μm) is demonstrated.

© 1984 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 371.
  2. R. M. A. Azzam, A. R. M. Zaghloul, N. M. Bashara, “Ellipsometric Function of a Film-Substrate System: Applications to the Design of Reflection-Type Optical Devices and to Ellipsometry,” J. Opt. Soc. Am. 65, 252 (1975).
    [CrossRef]
  3. A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “Design of Film–Substrate Single-Reflection Retarders,” J. Opt. Soc. Am. 65, 1043 (1975).
    [CrossRef]
  4. A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “An Angle-of-Incidence Tunable, SiO2–Si (Film–Substrate) Reflection Retarder for the UV Mercury Line λ = 2537 Å,” Opt. Commun. 14, 260 (1975).
    [CrossRef]
  5. A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “SiO2–Si Film–Substrate Reflection Retarders for Different Mercury Spectral Lines,” Opt. Eng. 17, 180 (1978).
    [CrossRef]
  6. S. Kawabata, M. Suzuki, “MgF2–Ag Tunable Reflection Retarder,” Appl. Opt. 19, 484 (1980).
    [CrossRef] [PubMed]
  7. R. M. A. Azzam, M. E. R. Khan, “Single-Reflection Film–Substrate Halfwave Retarders with Nearly Stationary Reflection Properties over a Wide Range of Incidence Angles,” J. Opt. Soc. Am. 73, 160 (1983).
    [CrossRef]
  8. W. H. Southwell, “Multilayer Coatings Producing 90° Phase Change,” Appl. Opt. 18, 1875 (1979).
    [CrossRef] [PubMed]
  9. J. H. Apfel, “Graphical Method to Design Multilayer Phase Retarders,” Appl. Opt. 20, 1024 (1981).
    [CrossRef] [PubMed]
  10. J. H. Apfel, “Phase Retardance of Periodic Multilayer Mirrors,” Appl. Opt. 21, 733 (1982).
    [CrossRef] [PubMed]
  11. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.3.
  12. R. M. A. Azzam, M. E. R. Khan, “Polarization-Preserving Single-Layer-Coated Beam Displacers and Axicons,” Appl. Opt. 21, 3314 (1982).
    [CrossRef] [PubMed]
  13. G. W. Stagg, A. H. El-Abiad, Computer Methods in Power System Analysis (McGraw-Hill, New York, 1968), p. 249.

1983 (1)

1982 (2)

1981 (1)

1980 (1)

1979 (1)

1978 (1)

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “SiO2–Si Film–Substrate Reflection Retarders for Different Mercury Spectral Lines,” Opt. Eng. 17, 180 (1978).
[CrossRef]

1975 (3)

Apfel, J. H.

Azzam, R. M. A.

R. M. A. Azzam, M. E. R. Khan, “Single-Reflection Film–Substrate Halfwave Retarders with Nearly Stationary Reflection Properties over a Wide Range of Incidence Angles,” J. Opt. Soc. Am. 73, 160 (1983).
[CrossRef]

R. M. A. Azzam, M. E. R. Khan, “Polarization-Preserving Single-Layer-Coated Beam Displacers and Axicons,” Appl. Opt. 21, 3314 (1982).
[CrossRef] [PubMed]

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “SiO2–Si Film–Substrate Reflection Retarders for Different Mercury Spectral Lines,” Opt. Eng. 17, 180 (1978).
[CrossRef]

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “An Angle-of-Incidence Tunable, SiO2–Si (Film–Substrate) Reflection Retarder for the UV Mercury Line λ = 2537 Å,” Opt. Commun. 14, 260 (1975).
[CrossRef]

R. M. A. Azzam, A. R. M. Zaghloul, N. M. Bashara, “Ellipsometric Function of a Film-Substrate System: Applications to the Design of Reflection-Type Optical Devices and to Ellipsometry,” J. Opt. Soc. Am. 65, 252 (1975).
[CrossRef]

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “Design of Film–Substrate Single-Reflection Retarders,” J. Opt. Soc. Am. 65, 1043 (1975).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 371.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.3.

Bashara, N. M.

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “SiO2–Si Film–Substrate Reflection Retarders for Different Mercury Spectral Lines,” Opt. Eng. 17, 180 (1978).
[CrossRef]

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “An Angle-of-Incidence Tunable, SiO2–Si (Film–Substrate) Reflection Retarder for the UV Mercury Line λ = 2537 Å,” Opt. Commun. 14, 260 (1975).
[CrossRef]

R. M. A. Azzam, A. R. M. Zaghloul, N. M. Bashara, “Ellipsometric Function of a Film-Substrate System: Applications to the Design of Reflection-Type Optical Devices and to Ellipsometry,” J. Opt. Soc. Am. 65, 252 (1975).
[CrossRef]

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “Design of Film–Substrate Single-Reflection Retarders,” J. Opt. Soc. Am. 65, 1043 (1975).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 371.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.3.

El-Abiad, A. H.

G. W. Stagg, A. H. El-Abiad, Computer Methods in Power System Analysis (McGraw-Hill, New York, 1968), p. 249.

Kawabata, S.

Khan, M. E. R.

Southwell, W. H.

Stagg, G. W.

G. W. Stagg, A. H. El-Abiad, Computer Methods in Power System Analysis (McGraw-Hill, New York, 1968), p. 249.

Suzuki, M.

Zaghloul, A. R. M.

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “SiO2–Si Film–Substrate Reflection Retarders for Different Mercury Spectral Lines,” Opt. Eng. 17, 180 (1978).
[CrossRef]

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “An Angle-of-Incidence Tunable, SiO2–Si (Film–Substrate) Reflection Retarder for the UV Mercury Line λ = 2537 Å,” Opt. Commun. 14, 260 (1975).
[CrossRef]

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “Design of Film–Substrate Single-Reflection Retarders,” J. Opt. Soc. Am. 65, 1043 (1975).
[CrossRef]

R. M. A. Azzam, A. R. M. Zaghloul, N. M. Bashara, “Ellipsometric Function of a Film-Substrate System: Applications to the Design of Reflection-Type Optical Devices and to Ellipsometry,” J. Opt. Soc. Am. 65, 252 (1975).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “An Angle-of-Incidence Tunable, SiO2–Si (Film–Substrate) Reflection Retarder for the UV Mercury Line λ = 2537 Å,” Opt. Commun. 14, 260 (1975).
[CrossRef]

Opt. Eng. (1)

A. R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “SiO2–Si Film–Substrate Reflection Retarders for Different Mercury Spectral Lines,” Opt. Eng. 17, 180 (1978).
[CrossRef]

Other (3)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.3.

G. W. Stagg, A. H. El-Abiad, Computer Methods in Power System Analysis (McGraw-Hill, New York, 1968), p. 249.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 371.

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Figures (15)

Fig. 1
Fig. 1

Three-mirror reflection system.

Fig. 2
Fig. 2

Normalized film thickness ζ1 required for HWR (|ρn| = 1, Δn = ±180°) vs angle of incidence ϕ1 (deg) in a ZnS–Ag three-reflection system at λ = 10.6 μm.

Fig. 3
Fig. 3

Same as in Fig. 2 for the associated normalized film tthickness ζ2.

Fig. 4
Fig. 4

Net intensity reflectances pn = sn associated with the design solution pairs (ζ1,ζ2) of Figs. 2 and 3.

Fig. 5
Fig. 5

Magnitude error (|ρn| − 1) and phase error [Δn − (180°)] caused by angle-of-incidence Δϕ1 = Δϕ2 = Δϕ3 = ±0.5° for solution pair B at ϕ1 = 80° of Table I.

Fig. 6
Fig. 6

Same as in Fig. 5 for Δd1 = Δd2 = ±1-nm changes of actual film thickness d1 and d2.

Fig. 7
Fig. 7

Same as in Fig. 5 for wavelength changes in the range 10 μm ≤ λ ≤ 11 μm.

Fig. 8
Fig. 8

Normalized film thickness ζ1 required for QWR (|ρn| = 1, Δn = 90°) vs angle of incidence ϕ1 (deg) in a ZnS–Ag three-reflection system at λ = 10.6 μm.

Fig. 9
Fig. 9

Same as in Fig. 2 for the associated normalized film thickness ζ2.

Fig. 10
Fig. 10

Net intensity reflectances pn = sn associated with the design solution pairs (ζ1,ζ2) of Figs. 8 and 9.

Fig. 11
Fig. 11

Magnitude error (|ρn| − 1) and phase error (Δn − 90°) caused by wavelength changes in the range 10 μm ≤ λ ≤ 11 μm for solution pair B at ϕ1 = 75° of Table III.

Fig. 12
Fig. 12

Normalized film thickness ζ1 required for QWR (|ρn| = 1, Δn = −90°) vs angle of incidence ϕ1 (deg) in a ZnS–Ag three-reflection system at λ = 10.6 μm.

Fig. 13
Fig. 13

Same as in Fig. 12 for the associated normalized film thickness ζ2.

Fig. 14
Fig. 14

Net intensity reflectances pn = sn associated with the design solution pairs (ζ1,ζ2) of Figs. 12 and 13.

Fig. 15
Fig. 15

Magnitude error (|ρn| − 1) and phase error [Δn − (−90°)] caused by wavelength changes in the range 10 μm ≤ λ ≤ 11 μm for solution pair A at ϕ1 = 75° of Table V.

Tables (6)

Tables Icon

Table I Summary of Design Results for ZnS–Ag Three-Reflection Halfwave Retarders (HWR: |ρn| = 1, Δn = δpnδsn = ±180°) at Two Angles of Incidence and for Wavelength λ = 10.6 μma

Tables Icon

Table II Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1 = Δd2 = ±1 nm), (II) Angle-of-Incidence Errors (Δϕ1 = Δϕ2 = Δϕ3 = ±0.5°) to the HWR Designs of Table I

Tables Icon

Table III Summary of Design Results for ZnS–Ag Three-Reflection Quarterwave Retarders with p Fast Axis (QWR: |ρn| = 1, Δn = δpnδsn = 90°) at Three Angles of Incidence and for Wavelength λ = 10.6 μma

Tables Icon

Table IV Magnitude and Phase Errors Caused by Introducing(I) Film-Thickness Errors (Δd1 = Δd2 = ±1 nm), (II) Angle-of-incidence Errors (Δδ1 = Δϕ2 = Δϕ3 = ±0.5°) to the QWR Designs of Table III

Tables Icon

Table V Summary of Design Results for ZnS–Ag Three-Reflection Quarterwave Retarders with s Fast Axis (QWR: |ρn| = 1, Δn = δpnδsn = −90°) at Three Angles of Incidence and at Wavelength λ = 10.6 μm a

Tables Icon

Table VI Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1 = Δd2 = ±1 nm),(II) Angle-of-Incidence Errors (Δϕ1 = Δϕ2 = Δϕ3 = ±0.5°) to the QWR Designs of Table V

Equations (29)

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ϕ 3 = ϕ 1 ,
ϕ 2 = 2 ϕ 1 - 90 ° .
ρ = R p / R s ,
R ν = r 01 ν + r 12 ν exp ( - j 2 π ζ ) 1 + r 01 ν r 12 ν exp ( - j 2 π ζ ) , ν = p , s .
ζ = d / D ϕ ,
D ϕ = λ 2 ( N 1 2 - N 0 2 sin 2 ϕ ) - 1 / 2
ρ = f ( ϕ , ζ ) .
ρ n = ρ 1 ρ 2 ρ 3 = f 2 ( ϕ 1 , ζ 1 ) f ( ϕ 2 , ζ 2 ) .
ρ n = ρ n exp ( j Δ n ) .
ρ n = R p n / R s n ,
Δ n = δ p n - δ s n ,
f ( ϕ 2 , ζ 2 ) = g ( ϕ 1 , ζ 1 ) ,
g ( ϕ 1 , ζ 1 ) = ρ n / f 2 ( ϕ 1 , ζ 1 ) .
f r ( ϕ 2 , ζ 2 ) = g r ( ϕ 1 , ζ 1 ) ,
f i ( ϕ 2 , ζ 2 ) = g i ( ϕ 1 , ζ 1 ) ,
f = f r + j f i ,             g = g r + j g i .
0 ζ 1 , 2 < 1.
f r ( ϕ 2 , ζ 2 0 + Δ ζ 2 ) = g r ( ϕ 1 , ζ 1 0 + Δ ζ 1 ) ,
f i ( ϕ 2 , ζ 2 0 + Δ ζ 2 ) = g i ( ϕ 1 , ζ 1 0 + Δ ζ 1 ) .
f r ( ϕ 2 , ζ 2 0 ) Δ ζ 2 - g r ( ϕ 1 , ζ 1 0 ) Δ ζ 1 = g r ( ϕ 1 , ζ 1 0 ) - f r ( ϕ 2 , ζ 2 0 ) ,
f i ( ϕ 2 , ζ 2 0 ) Δ ζ 2 - g i ( ϕ 1 ζ 1 0 ) Δ ζ 1 = g i ( ϕ 1 , ζ 1 0 ) - f i ( ϕ 2 , ζ 2 0 ) .
f = ρ = ( R p / R s ) - ( R p R s / R s 2 ) ,
R ν = - j 2 π exp ( - j 2 π ζ ) r 12 ν ( 1 - r 01 ν 2 ) [ 1 + r 01 ν r 12 ν exp ( - j 2 π ζ ) ] 2 , ν = p , s ,
g = - 2 ρ n f / f 3 ,
f r = Re f , f i = Im f ; g r = Re g , g i = Im g .
( d 1 , d 2 ) = ( ζ 1 D ϕ 1 , ζ 2 D ϕ 2 ) .
R ν n = R ν 1 2 R ν 2 2 R ν 3 2 ,             ν = p or s ,
R p m = R s n .
magnitude error = ρ n - 1 , phase error = Δ n - Δ n d ,

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